Properties

Label 69696.cd
Number of curves $4$
Conductor $69696$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cd1")
 
E.isogeny_class()
 

Elliptic curves in class 69696.cd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69696.cd1 69696gq4 \([0, 0, 0, -3068076, -2068459184]\) \(37736227588/33\) \(2793042278940672\) \([2]\) \(1474560\) \(2.2641\)  
69696.cd2 69696gq3 \([0, 0, 0, -454476, 72811024]\) \(122657188/43923\) \(3717539273270034432\) \([2]\) \(1474560\) \(2.2641\)  
69696.cd3 69696gq2 \([0, 0, 0, -193116, -31837520]\) \(37642192/1089\) \(23042598801260544\) \([2, 2]\) \(737280\) \(1.9176\)  
69696.cd4 69696gq1 \([0, 0, 0, 2904, -1650440]\) \(2048/891\) \(-1178314711428096\) \([2]\) \(368640\) \(1.5710\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 69696.cd have rank \(0\).

Complex multiplication

The elliptic curves in class 69696.cd do not have complex multiplication.

Modular form 69696.2.a.cd

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} + 4 q^{7} + 6 q^{13} + 6 q^{17} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.